Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions

Abstract

We develop a method of constructing percolation clusters that allows us to build very large clusters using very little computer memory by limiting the maximum number of sites for which we maintain state information to a number of the order of the number of sites in the largest chemical shell of the cluster being created. The memory required to grow a cluster of mass s is of the order of s θ bytes where θ ranges from 0.4 for 2-dimensional lattices to 0.5 for 6-(or higher)-dimensional lattices. We use this method to estimate d min , the exponent relating the minimum path ℓ to the Euclidean distance r, for 4D and 5D hypercubic lattices. Analyzing both site and bond percolation, we find d min = 1.607 ± 0.005 (4D) and d min = 1.812 ± 0.006 (5D). In order to determine d min to high precision, and without bias, it was necessary to first find precise values for the percolation threshold, pc: pc = 0.196889 ± 0.000003 (4D) and pc = 0.14081 ± 0.00001 (5D) for site and pc = 0.160130±0.000003 (4D) and pc = 0.118174±0.000004 (5D) for bond percolation. We also calculate the Fisher exponent, τ , determined in the course of calculating the values of pc: τ = 2.313 ± 0.003 (4D) and τ = 2.412 ± 0.004 (5D).

High-temperature series expansions for theq-state Potts model on a hypercubic lattice and critical properties of percolation

“…MC (Lorenz and Ziff, 1998a) 0.248 812 6(5) MCS (Ballesteros et al, 1999) 0.4522(8) 2.18906(6) 1.7933(85) MCS (Deng and Blöte, 2005) 0.4539(3) 2.18925(5) 1.7862(30) present work 0.248 91(10) 1.804(5) HTS (Adler et al, 1990) 0.160 05(15) 1.435(15) MCS (Ballesteros et al, 1997) 1.44(2) 4 MC (Paul et al, 2001) 0.160 130(3) 2.313(3) MC (Grassberger, 2003) 0.160 131 4(13) present work 0.160 08(10) 1.435(5) HTS (Adler et al, 1990) 0.118 19(4) 1.185 (5) 5 MC (Paul et al, 2001) 0.118 174(4) 2.412(4) MC (Grassberger, 2003) 0.118 172(1) present work 0.118 170(5) 1.178(2) RG (Essam et al, 1978) 1 2 5 2…”

Concentration gradient, diffusion, and flow through open porous medium near percolation threshold via computer simulations

“…Particularly at p = 0.312, the lowest porosity we simulated near percolation threshold (p c ≃ 0.311608) [7], it was difficult to reach long time power-law behavior with the larger samples. Therefore, we performed our simulation on a smaller sample (30 3 ) for 5 × 10 6 time steps.…”

“…How does the flux-rate depend on porosity? Addressing these questions becomes somewhat difficult with respect to solving the diffusion equation (1) particularly near percolation threshold [7][8][9] where the percolating pores are highly ramified; the boundary conditions involved with the pore space become prohibitively large to solve the diffusion equation numerically. We consider an interacting lattice gas to model the fluid in order to address these questions in a percolating porous matrix.…”

“…In a percolating system at porosity (the fraction of pore sites) above threshold, p > p c , we have ν 1 = 1/2. At the percolation threshold [7][8][9], on the other hand, the random walk motion becomes anomalous diffusion with ν 1 ≃ 0.2 in three dimensions and ν 1 ≃ 0.3 in two dimensions [10].…”

Red-bond exponents of the critical and the tricritical Ising model in three dimensions

“…In the percolation theory, these exponents have received considerable attention and are considered to be of some physical relevance. For the q → 1 Potts model, the red-bond exponent y r just reduces to the thermal exponent y t , which is about 1.14(2) in three dimensions [48].…”